After what interval in degrees or radians do sine, cosine and tangent values repeat? Between $0$ to $2π$, I have noticed that $\sin x$, $\cos x$ and $\tan x$ values repeat for different values of $x$.
For example, $\sin 30 = \sin 150$
What exactly is the interval between two successive values of $x$ such that the value of $\sin x$, $\cos x$ or $\tan x$ are equal?
 A: $\sin(x) = \sin(\pi - x)$, or, equivalently
$\sin(x) = \sin(180^o - x)$.  
$\sin(x) = \sin(2\pi n + x) \forall n \in \mathbb{Z}$, or, equivalently,
$\sin(x) = \sin(360^o n + x) \forall n \in \mathbb{Z}$.  
$\cos(x) = \cos(-x)$. Alternately,
$\cos(x) = \cos(360^o - x)$, or (equivalently to the latter)
$\cos(x) = \cos(2\pi - x)$.  
$\cos(x) = \cos(2\pi n + x) \forall n \in \mathbb{Z}$, or, equivalently
$\cos(x) = \cos(360^o n + x) \forall n \in \mathbb{Z}$.  
$\tan(x) = \tan(\pi + x)$, or, equivalently,
$\tan(x) = \tan(180^o + x)$.
$\tan(x) = \tan(\pi n + x) \forall n \in \mathbb{Z}$, or, equivalently,
$\tan(x) = \tan(180^o n + x) \forall n \in \mathbb{Z}$.  
A: $1:$
Let $\sin x=\sin y$
Applying $\sin C-\sin D=2\sin\frac{C-D}2\cos\frac{C+D}2,$
we have  $2\sin\frac{x-y}2\cos\frac{x+y}2=0$
If $\sin\frac{x-y}2=0, \frac{x-y}2=n180^\circ\implies x-y=n180^\circ$ where $n$ is any integer
If $\cos\frac{x+y}2=0, \frac{x+y}2=(2m+1)90^\circ\implies x+y=(2m+1)180^\circ$ where $m$ is any integer
$2:$
Apply $\cos C-\cos D=-2\sin\frac{C-D}2\sin\frac{C+D}2,$
$3:$
$\displaystyle\tan x=\tan y\iff \frac{\sin x}{\cos x}=\frac{\sin y}{\cos y}$
$$\implies \sin x\cos y-\cos x\sin y=0\implies \sin(x-y)=0$$
$\implies x-y=r180^\circ$ where $r$ is any integer
